document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207035>Question 1207035</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The number of births at the local hospital has a Poisson distribution with an average of 48 per day.\r<BR>\n" );
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document.write( "What is the probability distribution for the daily number of births at this hospital?<BR>\n" );
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document.write( "What is the probability distribution for the number of hourly births?<BR>\n" );
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document.write( "(c)<BR>\n" );
document.write( "What is the probability that there are fewer than 3 births in a given hour? (Round your answer to three decimal places.)\r<BR>\n" );
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document.write( "(d)<BR>\n" );
document.write( "Within what interval does Tchebysheff's Theorem suggest you would expect to find the number of hourly births at least 89% of the time? (Round your answer up to the nearest whole number.)<BR>\n" );
document.write( "interval (a,b) =  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #844933 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=844933\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you can use the poisson distribution calculator at <a  rel=nofollow HREF= \"\" target = \"_blank\"></a> to solve parts (a) through (c).<BR>\n" );
document.write( "an alternative is to use the poisson distribution table.\r<BR>\n" );
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document.write( "here's a reference on the poisson distribution itself.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.youtube.com/watch?v=zA7fp2s7FlM\" target = \"_blank\">https://www.youtube.com/watch?v=zA7fp2s7FlM</a>\r<BR>\n" );
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document.write( "here's a link to the calculator.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://stattrek.com/online-calculator/poisson#google_vignette\" target = \"_blank\">https://stattrek.com/online-calculator/poisson#google_vignette</a>\r<BR>\n" );
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document.write( "here are the results from using the calculator.\r<BR>\n" );
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document.write( "you have to set the time interval to be consistent.<BR>\n" );
document.write( "you were given 48 births in a 24 hour period.<BR>\n" );
document.write( "since you want number per hour, you have to set the average to per hoursly.<BR>\n" );
document.write( "48 per 24 hours is equivalent to 2 per hour because the hospital is open 24 hours a day.<BR>\n" );
document.write( "so the average  is 2 per hour and the variable you are looking for is 3.<BR>\n" );
document.write( "you enter 2 for the average and 3 for the number you are looking for and the calculator gives you a bunch of statistics.<BR>\n" );
document.write( "since you are looking for x < 3, you choose that.<BR>\n" );
document.write( "you get p(x<3) = 0.67668.\r<BR>\n" );
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document.write( "part (a) answer is 48.<BR>\n" );
document.write( "part (b) answer is 2.<BR>\n" );
document.write( "part (c) answer is .67668.\r<BR>\n" );
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document.write( "you can also use the poisson dictribution table for cumulative results.<BR>\n" );
document.write( "one such table can be found at <a  rel=nofollow HREF= \"https://ghaidab.weebly.com/uploads/1/1/2/9/11294132/poisson_cdf_table.pdf\" target = \"_blank\">https://ghaidab.weebly.com/uploads/1/1/2/9/11294132/poisson_cdf_table.pdf</a>\r<BR>\n" );
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document.write( "lambda is the mean of the distribution, so for this problem, you would look for a lambda of 2 and a value of x as 2.<BR>\n" );
document.write( "you are looking for p(x) < 3 which is the same as p(x) <= 2.<BR>\n" );
document.write( "p(x) <= is what the table is set up to look for.<BR>\n" );
document.write( "you should find that p(x <= 2) = .6767 from the table.<BR>\n" );
document.write( "this agrees with what the calculator provided as p(x) < 3.\r<BR>\n" );
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document.write( "if you wanted to find p(x) > 3, you would look for 1 - p(x) <= 3.<BR>\n" );
document.write( "that would be equal to 1 minus p(x <= 3) = 1 - .0.85712 = .14288.<BR>\n" );
document.write( "if you're using the calculator, it tells you othat straight away.<BR>\n" );
document.write( "if you're using the table, you look for p(x) <= 3 and take 1 minus that.\r<BR>\n" );
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document.write( "the calculator goes to 5 decimal places.<BR>\n" );
document.write( "the table goes to 4.\r<BR>\n" );
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document.write( "for part (d), you need to use chebyshev's theorem.<BR>\n" );
document.write( "that's the way the internet spells it, i believe.<BR>\n" );
document.write( "that theorem states that the smallest probability of any type of distribution is given by the equation of p(|x-m|>=k*s) <= 1 - 1/k^2.<BR>\n" );
document.write( "this says that the probability that the variable value minus the mean is greater than or equal to k * the standard deviation is smaller than or equal to 1 - 1/k^2.\r<BR>\n" );
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document.write( "k is the number of standard deviations above the mean.<BR>\n" );
document.write( "the mean is assumed to be in the middle of the range, so the standard deviation is the same whether above or below the mean.<BR>\n" );
document.write( "the absolute value symbol takes care of that by making (x-s) always positive.\r<BR>\n" );
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document.write( "in your problem, mean is 2 and variance is also 2, since this is a property of the poisson distribution.<BR>\n" );
document.write( "standard deviation is therefore sqrt(2).\r<BR>\n" );
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document.write( "at least 89% of the time means the probability that it is greater than or equal to 89% = .89<BR>\n" );
document.write( ".89 is the minimum proportion.<BR>\n" );
document.write( "the formula is p(|x-m| >= k*s) >= 1 - 1/k^2 = .89<BR>\n" );
document.write( "add .89 to both sides of the equation abd add 1/k^2 to both sides of the equation to get 1.89 = 1/k^2.<BR>\n" );
document.write( "solve for k^2 to get k^2 = 1/1.89.<BR>\n" );
document.write( "solve for k to get k = sqrt(1/1.89) = .72739<BR>\n" );
document.write( "the interpretatio9n of that would be that:<BR>\n" );
document.write( "the minimum percentage of values that are .72739 * the standard deviation about the mean is equal to .89.\r<BR>\n" );
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document.write( "here's a reference on chebyshev's formula.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://statisticsbyjim.com/basics/chebyshevs-theorem-in-statistics/\" target = \"_blank\">https://statisticsbyjim.com/basics/chebyshevs-theorem-in-statistics/</a>\r<BR>\n" );
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document.write( "the reference states that the maximum proportion = 1/k^2 and the minimum proportion is 1 - 1/k^2.\r<BR>\n" );
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document.write( "we used the minimum proportion to solve your problem.\r<BR>\n" );
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